Question

Calculate the wave length for several examples of sinusoidal electromagnetic radiation:

radio,$\mathbf{1000}\mathbf{}\mathit{k}\mathit{H}\mathit{z}\mathbf{,}\mathbf{}\mathit{\lambda }\mathbf{=}\mathbf{}\mathbf{?}$

television,$\mathbf{100}\mathbf{}\mathit{M}\mathit{H}\mathit{z}\mathbf{,}\mathbf{}\mathit{\lambda }\mathbf{=}\mathbf{}\mathbf{?}$

red light,$\mathbf{4}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{}\mathit{H}\mathit{z}\mathbf{,}\mathbf{}\mathit{\lambda }\mathbf{=}\mathbf{}\mathbf{?}$

blue light,$\mathbf{7}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{}\mathit{H}\mathit{z}\mathbf{,}\mathbf{}\mathit{\lambda }\mathbf{=}\mathbf{}\mathbf{?}$

(Note for comparison that an atomic diameter is about $\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{10}}\mathbf{}\mathit{m}\mathbf{,}$).

Short answer

The wavelength of the radio, television, red light and blue light are $300\text{m,}3\text{m,}697.6\text{nm,and}400\text{nm}$respectively.

Step-by-step solution

Identification of given data

Frequency of the radio,$f=1000\text{kHz}$

Frequency of the television,$f=100\text{MHz}$

Frequency of the red light,$f=4.3\times {10}^{14}\text{Hz}$

Frequency of the blue light,$f=7.5\times {10}^{14}\text{Hz}$

Determine the formula to calculate the wavelength of the radio, television, red light and blue light.

The wavelength of the frequency and speed of the light is defined as the ratio of the light of the speed and frequency.

$\mathit{\lambda }\mathbf{=}\frac{\mathbf{v}}{\mathbf{f}}$ …(i)

Here,$\mathit{\nu }$ is the speed of the light.

Calculate the wavelength of the radio, television, red light and blue light.

Calculate the wavelength of the radio.

Substitute$1000\text{kHz}$ for f and$3\times {10}^{8}m/s$ for$\nu$ into equation (i).

$$\begin{gathered} \lambda =\frac{3\times {10}^{8}m/s}{1000\times {10}^3\text{Hz}} \\ =\frac{3\times {10}^{8}m/s}{{10}^{6}1/s} \\ =3\times {10}^2\text{m} \\ =300\text{m} \end{gathered}$$

Therefore, the wavelength of the radio is $300\text{m}$.

Calculate the wavelength of the television.

Substitute $100\text{MHzforf}$and $3\times {10}^{8}m/sfor\nu$into equation (i).

$$\begin{gathered} d\lambda =\frac{3\times {10}^{8}m/s}{100\times {10}^6\text{Hz}} \\ =\frac{3\times {10}^{8}m/s}{{10}^{8}1/s} \\ =3\text{m} \end{gathered}$$

Therefore, the wavelength of the television is $3m$.

Calculate the wavelength of the red light.

Substitute $4.3\times {10}^{14}\text{Hzforf}$and $3\times {10}^{8}m/sfor\nu$into equation (i).

$$\begin{gathered} \lambda =\frac{3\times {10}^{8}m/s}{\text{4.3}\times {\text{10}}^{14}\text{Hz}} \\ =\frac{3\times {10}^{-6}m/s}{\text{4.3}1/s} \\ =0.6976\times {10}^{-6}\text{m} \\ =697.6\text{nm} \end{gathered}$$

Therefore, the wavelength of the red light is $697.6nm$.

Calculate the wavelength of the red light.

Substitute $7.5\times {10}^{14}\text{Hzforf}$and $3\times {10}^{8}m/sfor\nu$into equation (i).

$$\begin{gathered} \lambda =\frac{3\times {10}^{8}m/s}{7.5\times {\text{10}}^{14}\text{Hz}} \\ =\frac{3\times {10}^{-6}m/s}{7.51/s} \\ =0.4\times {10}^{-6}\text{m} \\ =400\text{nm} \end{gathered}$$

Therefore, the wavelength of the blue light is $400nm$.

Hence the wavelength of the radio, television, red light, and blue light are $300m,3m,697.6nm,and400nm$respectively.